<?xml version="1.0" encoding="utf-8"?>
<InstanceReport xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:xsd="http://www.w3.org/2001/XMLSchema">
  <Version>1.0.0.3</Version>
  <hasSegments>false</hasSegments>
  <ReportName>Warrant and Public Offerings</ReportName>
  <RoundingOption />
  <Columns>
    <Column>
      <LabelColumn>false</LabelColumn>
      <Id>1</Id>
      <Labels>
        <Label Id="1" Label="12 Months Ended" />
        <Label Id="2" Label="Dec. 31, 2009" />
        <Label Id="4" Label="USD / shares" />
      </Labels>
      <CurrencySymbol>$</CurrencySymbol>
      <hasSegments>false</hasSegments>
      <hasScenarios>false</hasScenarios>
      <Segments />
      <Scenarios />
      <Units>
        <Unit>
          <UnitID>Shares</UnitID>
          <UnitType>Standard</UnitType>
          <StandardMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema>
            <MeasureValue>shares</MeasureValue>
            <MeasureNamespace>xbrli</MeasureNamespace>
          </StandardMeasure>
          <Scale>0</Scale>
        </Unit>
        <Unit>
          <UnitID>USD</UnitID>
          <UnitType>Standard</UnitType>
          <StandardMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema>
            <MeasureValue>USD</MeasureValue>
            <MeasureNamespace>iso4217</MeasureNamespace>
          </StandardMeasure>
          <Scale>0</Scale>
        </Unit>
        <Unit>
          <UnitID>USDperShare</UnitID>
          <UnitType>Divide</UnitType>
          <NumeratorMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/iso4217</MeasureSchema>
            <MeasureValue>USD</MeasureValue>
            <MeasureNamespace>iso4217</MeasureNamespace>
          </NumeratorMeasure>
          <DenominatorMeasure>
            <MeasureSchema>http://www.xbrl.org/2003/instance</MeasureSchema>
            <MeasureValue>shares</MeasureValue>
            <MeasureNamespace>xbrli</MeasureNamespace>
          </DenominatorMeasure>
          <Scale>0</Scale>
        </Unit>
      </Units>
    </Column>
  </Columns>
  <Rows>
    <Row>
      <Id>2</Id>
      <Label>Warrant and Public Offerings [Abstract]</Label>
      <Level>0</Level>
      <ElementName>apei_WarrantAndPublicOfferingsAbstract</ElementName>
      <ElementPrefix>apei</ElementPrefix>
      <IsBaseElement>false</IsBaseElement>
      <BalanceType>na</BalanceType>
      <PeriodType>duration</PeriodType>
      <ElementDataType>string</ElementDataType>
      <ShortDefinition>No definition available.</ShortDefinition>
      <IsReportTitle>false</IsReportTitle>
      <IsSegmentTitle>false</IsSegmentTitle>
      <IsSubReportEnd>false</IsSubReportEnd>
      <IsCalendarTitle>false</IsCalendarTitle>
      <IsTuple>false</IsTuple>
      <IsAbstractGroupTitle>true</IsAbstractGroupTitle>
      <IsBeginningBalance>false</IsBeginningBalance>
      <IsEndingBalance>false</IsEndingBalance>
      <IsEPS>false</IsEPS>
      <Cells>
        <Cell>
          <Id>1</Id>
          <ShowCurrencySymbol>false</ShowCurrencySymbol>
          <IsNumeric>false</IsNumeric>
          <NumericAmount>0</NumericAmount>
          <RoundedNumericAmount>0</RoundedNumericAmount>
          <NonNumbericText />
          <NonNumericTextHeader />
          <FootnoteIndexer />
          <hasSegments>false</hasSegments>
          <hasScenarios>false</hasScenarios>
        </Cell>
      </Cells>
      <ElementDefenition>No definition available.</ElementDefenition>
      <IsTotalLabel>false</IsTotalLabel>
    </Row>
    <Row>
      <Id>3</Id>
      <Label>Warrant and Public Offerings [Text Block]</Label>
      <Level>1</Level>
      <ElementName>apei_WarrantAndPublicOfferingsDisclosureTextBlock</ElementName>
      <ElementPrefix>apei</ElementPrefix>
      <IsBaseElement>false</IsBaseElement>
      <BalanceType>na</BalanceType>
      <PeriodType>duration</PeriodType>
      <ElementDataType>string</ElementDataType>
      <ShortDefinition>No definition available.</ShortDefinition>
      <IsReportTitle>false</IsReportTitle>
      <IsSegmentTitle>false</IsSegmentTitle>
      <IsSubReportEnd>false</IsSubReportEnd>
      <IsCalendarTitle>false</IsCalendarTitle>
      <IsTuple>false</IsTuple>
      <IsAbstractGroupTitle>false</IsAbstractGroupTitle>
      <IsBeginningBalance>false</IsBeginningBalance>
      <IsEndingBalance>false</IsEndingBalance>
      <IsEPS>false</IsEPS>
      <Cells>
        <Cell>
          <Id>1</Id>
          <ShowCurrencySymbol>false</ShowCurrencySymbol>
          <IsNumeric>false</IsNumeric>
          <NumericAmount>0</NumericAmount>
          <RoundedNumericAmount>0</RoundedNumericAmount>
          <NonNumbericText>&lt;div align="left"&gt;
&lt;table cellpadding="0" width="100%" style="FONT-SIZE: 10pt; FONT-FAMILY: times new roman" cellspacing="0"&gt;
&lt;tr&gt;
&lt;td width="10%" valign="top" align="left"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0.8pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-WEIGHT: bold; FONT-SIZE: 10pt; FONT-FAMILY: times new roman"&gt;Note&amp;#160;8.&amp;#160;&amp;#160;&lt;/font&gt;&lt;/div&gt;&lt;/td&gt;
&lt;td width="90%" valign="top" align="left"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0.75pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-WEIGHT: bold; FONT-SIZE: 10pt; FONT-FAMILY: times new roman"&gt;Warrant and Public Offerings&lt;/font&gt;&lt;/div&gt;&lt;/td&gt;&lt;/tr&gt;&lt;/table&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="justify"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;&amp;#160;&lt;/font&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-WEIGHT: bold; FONT-SIZE: 10pt; FONT-STYLE: italic; FONT-FAMILY: Times New Roman"&gt;Warrant&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="justify"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;&amp;#160;&lt;/font&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 27pt; MARGIN-RIGHT: 0pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;In connection with an August 2002 equity offering, the Company issued a warrant (the &amp;#8220;Warrant&amp;#8221;) to a third party placement agent for its service in arranging and negotiating the offering. In August 2005, in connection with a subsequent financing the Company entered into an agreement to exchange the Warrant into a warrant to purchase 155,815&amp;#160;shares of Class&amp;#160;A common stock. The warrant was exercised in October 2007 at $4.62 per share. Upon exercise, the excess tax benefit was $718,000 and the intrinsic value was $1,772,000.&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="justify"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;&amp;#160;&lt;/font&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-WEIGHT: bold; FONT-SIZE: 10pt; FONT-STYLE: italic; FONT-FAMILY: Times New Roman"&gt;Initial Public Offering&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="justify"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;&amp;#160;&lt;/font&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 27pt; MARGIN-RIGHT: 0pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;On August&amp;#160;7, 2007, the Company filed a Registration Statement on Form&amp;#160;S-1 (Registration No.&amp;#160;333-145185) for its initial public offering, which was completed on November&amp;#160;14, 2007.&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="justify"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;&amp;#160;&lt;/font&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 27pt; MARGIN-RIGHT: 0pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;In the initial public offering, the Company sold 5,390,625&amp;#160;shares of common stock at a price to the public of $20.00 per share, before underwriting discounts and commissions. The sale of the shares included the exercise in full of the underwriters&amp;#8217; option to purchase up to an additional 703,125&amp;#160;shares at the initial public offering price to cover over-allotments. Net proceeds to the Company were approximately $100.3&amp;#160;million, after deducting underwriting discounts and commissions and before offering expenses. In connection with the closing of the initial public offering, all of the Class&amp;#160;A common stock was converted into shares of common stock on a 1 for 1 basis. The total number of shares of all classes of stock that the Company has the authority to issue is 110,000,000, of which 100,000,000 of such shares are common stock having a par value of $.01 per share and 10,000,000 of such shares are Preferred Stock, having a par value of $.01 per share.&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="justify"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;&amp;#160;&lt;/font&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 27pt; MARGIN-RIGHT: 0pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;On November&amp;#160;8, 2007, the Company declared a special distribution in the amount of $93,750,000 or $7.63 per share of common stock and Class&amp;#160;A common stock, payable upon the completion of the initial public offering to stockholders of record immediately prior to the completion of the offering. The Company used proceeds from the initial public offering to pay the special distribution. Shares of common stock issuable upon the exercise of outstanding stock options issued under prior plans were increased by 350,160&amp;#160;shares as a result of an equitable antidilution adjustment triggered by the special distribution.&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt"&gt;
&lt;div style="DISPLAY: block; TEXT-INDENT: 0pt"&gt;&lt;br/&gt;
&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-WEIGHT: bold; FONT-SIZE: 10pt; FONT-STYLE: italic; FONT-FAMILY: Times New Roman"&gt;Public Offerings&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="justify"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;&amp;#160;&lt;/font&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 27pt; MARGIN-RIGHT: 0pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;On January&amp;#160;25, 2008, APEI filed a Registration Statement on Form&amp;#160;S-1 (Registration No.&amp;#160;333-148851) for a public offering, which was completed on February&amp;#160;19, 2008. In the offering 3,744,500&amp;#160;shares were sold, consisting of 25,000&amp;#160;shares sold by the Company and 3,719,500&amp;#160;shares sold by certain stockholders of the Company. Total net proceeds to the Company were $167,000, after deducting underwriting discounts and commissions, and offering expenses. The Company did not receive any of the proceeds from the sale of common stock sold by the selling stockholders. Certain selling stockholders granted the underwriters a 30-day option to purchase up to an additional 500,175&amp;#160;shares at the public offering price to cover over-allotments. On February&amp;#160;27, 2008, the underwriters of the Company&amp;#8217;s public offering exercised their over-allotment option in full. The closing of the exercise of the over-allotment option occurred on March&amp;#160;3, 2008. The Company did not receive any of the proceeds from the sale of common stock held by the selling stockholders in the over-allotment option exercise.&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt" align="justify"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;&amp;#160;&lt;/font&gt;&lt;/div&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 0pt; MARGIN-RIGHT: 0pt"&gt;
&lt;div style="DISPLAY: block; MARGIN-LEFT: 0pt; TEXT-INDENT: 27pt; MARGIN-RIGHT: 0pt" align="left"&gt;&lt;font style="DISPLAY: inline; FONT-SIZE: 10pt; FONT-FAMILY: Times New Roman"&gt;On November&amp;#160;12, 2008, APEI filed a Registration Statement that was subsequently amended on Form&amp;#160;S-3 (Registration No.&amp;#160;333-155300) for a public offering, which was completed on December&amp;#160;12, 2008. In the offering 4,227,952&amp;#160;shares were sold consisting of 15,000&amp;#160;shares sold by the Company and 3,791,657&amp;#160;shares sold by certain stockholders of the Company. Total net proceeds to the Company were $52,280, after deducting underwriting discounts and commissions, and offering expenses. The Company did not receive any of the proceeds from the sale of common stock sold by the selling stockholders. Certain selling stockholders granted the underwriters a 30-day option to purchase up to an additional 421,295&amp;#160;shares at the public offering price to cover over-allotments. On December&amp;#160;9, 2008, the underwriters of the Company&amp;#8217;s public offering exercised their over-allotment option in full. The closing of the exercise of the over-allotment option occurred on December&amp;#160;9, 2008. The Company did not receive any of the proceeds from the sale of common stock held by the selling stockholders in the over-allotment option exercise.&lt;/font&gt;&lt;/div&gt;&lt;/div&gt;
</NonNumbericText>
          <NonNumericTextHeader>Note&amp;#160;8.&amp;#160;&amp;#160;

Warrant and Public Offerings
&amp;#160;

Warrant
&amp;#160;

In connection with an August 2002 equity offering, the Company issued a warrant</NonNumericTextHeader>
          <FootnoteIndexer />
          <hasSegments>false</hasSegments>
          <hasScenarios>false</hasScenarios>
        </Cell>
      </Cells>
      <ElementDefenition>No definition available.</ElementDefenition>
      <ElementReferences>No authoritative reference available.</ElementReferences>
      <IsTotalLabel>false</IsTotalLabel>
    </Row>
  </Rows>
  <Footnotes />
  <ComparabilityReport>false</ComparabilityReport>
  <NumberOfCols>1</NumberOfCols>
  <NumberOfRows>2</NumberOfRows>
  <HasScenarios>false</HasScenarios>
  <MonetaryRoundingLevel>UnKnown</MonetaryRoundingLevel>
  <SharesRoundingLevel>UnKnown</SharesRoundingLevel>
  <PerShareRoundingLevel>UnKnown</PerShareRoundingLevel>
  <HasPureData>false</HasPureData>
  <SharesShouldBeRounded>true</SharesShouldBeRounded>
</InstanceReport>
